Volume 5 (2022)
Duboscq, Romain;  Réveillac, Anthony
On a stochastic Hardy–Littlewood–Sobolev inequality with application to Strichartz estimates for a noisy dispersion
Annales Henri Lebesgue, Volume 5 (2022), pp. 263-274.

Metadata

DOI10.5802/ahl.122
Keywords Stochastic regularization, Stochastic Partial Differential Equations, Nonlinear Schrödinger equation, Hardy–Littlewood–Sobolev inequality

Abstract

In this paper, we investigate a stochastic Hardy–Littlewood–Sobolev inequality. Due to the non-homogenous nature of the potential in the inequality, we show that a constant proportional to the length of the interval appears on the right-hand-side. As a direct application, we derive local Strichartz estimates for randomly modulated dispersions and solve the Cauchy problem of the critical nonlinear Schrödinger equation.


References

[Agr01] Agrawal, Govind P. Applications of nonlinear fiber optics, Optics and photonics, Elsevier, 2001

[Agr07] Agrawal, Govind P. Nonlinear fiber optics, Academic Press Inc., 2007

[BBD15] Belaouar, Radoin; de Bouard, Anne; Debussche, Arnaud Numerical analysis of the nonlinear Schrödinger equation with white noise dispersion, Stoch. Partial Differ. Equ., Anal. Comput., Volume 3 (2015) no. 1, pp. 103-132 | Zbl

[BD99] de Bouard, Anne; Debussche, Arnaud A Stochastic Nonlinear Schrödinger Equation with Multiplicative Noise, Commun. Math. Phys., Volume 205 (1999) no. 1, pp. 161-181 | DOI | Zbl

[BD02] de Bouard, Anne; Debussche, Arnaud On the effect of a noise on the solutions of the focusing supercritical nonlinear Schrödinger equation, Probab. Theory Relat. Fields, Volume 123 (2002) no. 1, pp. 76-96 | DOI | Zbl

[BD05] de Bouard, Anne; Debussche, Arnaud Blow-up for the stochastic nonlinear Schrödinger equation with multiplicative noise, Ann. Probab., Volume 33 (2005) no. 3, pp. 1078-1110 | Zbl

[BD10] de Bouard, Anne; Debussche, Arnaud The nonlinear Schrödinger equation with white noise dispersion, J. Funct. Anal., Volume 259 (2010) no. 5, pp. 1300-1321 | DOI | Zbl

[Cat16] Catellier, Rémi Rough linear transport equation with an irregular drift, Stoch. Partial Differ. Equ., Anal. Comput., Volume 4 (2016) no. 3, pp. 477-534 | MR | Zbl

[Caz03] Cazenave, Thierry Semilinear schrödinger equations, Courant Lecture Notes in Mathematics, 10, American Mathematical Society; Courant Institute, 2003 | Zbl

[CG15] Chouk, Khalil; Gubinelli, Massimiliano Nonlinear PDEs with modulated dispersion I: Nonlinear Schrödinger equations, Commun. Partial Differ. Equations, Volume 40 (2015) no. 11, pp. 2047-2081 | DOI | Zbl

[CG16] Catellier, Rémi; Gubinelli, Massimiliano Averaging along irregular curves and regularisation of ODEs, Stochastic Processes Appl., Volume 126 (2016) no. 8, pp. 2323 -2366 | DOI | MR | Zbl

[CGM17] Chen, Min; Goubet, Olivier; Mammeri, Youcef Generalized regularized long wave equation with white noise dispersion, Stoch. Partial Differ. Equ., Anal. Comput., Volume 5 (2017) no. 3, pp. 319-342 | MR | Zbl

[DPFPR13] Da Prato, Giuseppe; Flandoli, Franco; Priola, Enrico; Röckner, Michael Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift, Ann. Probab., Volume 41 (2013) no. 5, pp. 3306-3344 | MR | Zbl

[DT11] Debussche, Arnaud; Tsutsumi, Yoshio 1D quintic nonlinear Schrödinger equation with white noise dispersion, J. Math. Pures Appl., Volume 96 (2011) no. 4, pp. 363-376 | DOI | Zbl

[ESY08] Erdös, Lázló; Salmhofer, Manfred; Yau, Horng-Tzer Quantum diffusion of the random Schrödinger evolution in the scaling limit, Acta Math., Volume 200 (2008) no. 2, p. 211 | DOI | Zbl

[EY00] Erdös, Lázló; Yau, Horng-Tzer Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation, Commun. Pure Appl. Math., Volume 53 (2000) no. 6, pp. 667-735 | DOI | Zbl

[FF13] Fedrizzi, Ennio; Flandoli, Franco Noise prevents singularities in linear transport equations, J. Funct. Anal., Volume 264 (2013) no. 6, pp. 1329-1354 | DOI | MR | Zbl

[FGP10] Flandoli, Franco; Gubinelli, Massimiliano; Priola, Enrico Well-posedness of the transport equation by stochastic perturbation, Invent. Math., Volume 180 (2010) no. 1, pp. 1-53 | DOI | MR | Zbl

[Fla11] Flandoli, Franco Random Perturbation of PDEs and Fluid Dynamic Models : École d’été de Probabilités de Saint-Flour XL–2010, Lecture Notes in Mathematics, 2015, Springer, 2011 | DOI | Zbl

[GS17] Gess, Benjamin; Souganidis, Panagiotis E. Long-Time Behavior, Invariant Measures, and Regularizing Effects for Stochastic Scalar Conservation Laws, Commun. Pure Appl. Math., Volume 70 (2017) no. 8, pp. 1562-1597 | DOI | MR | Zbl

[HL28] Hardy, Godfrey H.; Littlewood, John E. Some properties of fractional integrals. I., Math. Z., Volume 27 (1928) no. 1, pp. 565-606 | DOI | MR | Zbl

[HL31] Hardy, Godfrey H.; Littlewood, John E. Some properties of fractional integrals. II, Math. Z., Volume 34 (1931) no. 1, pp. 403-439 | DOI | MR | Zbl

[KR05] Krylov, Nikolaĭ V.; Röckner, Michael Strong solutions of stochastic equations with singular time dependent drift, Probab. Theory Relat. Fields, Volume 131 (2005) no. 2, pp. 154-196 | DOI | MR | Zbl

[KT98] Keel, Markus; Tao, Terence Endpoint strichartz estimates, Am. J. Math., Volume 120 (1998) no. 5, pp. 955-980 | DOI | MR | Zbl

[LL01] Lieb, Elliott H.; Loss, Michael Analysis, Graduate Studies in Mathematics, 14, American Mathematical Society, 2001 | Zbl

[Mar06] Marty, Renaud On a splitting scheme for the nonlinear Schrödinger equation in a random medium, Commun. Math. Sci., Volume 4 (2006) no. 4, pp. 679-705 | DOI | MR | Zbl

[Pri12] Priola, Enrico Pathwise uniqueness for singular SDEs driven by stable processes, Osaka J. Math., Volume 49 (2012) no. 2, pp. 421-447 | MR | Zbl

[Sob38] Sobolev, Sergeĭ L. On a theorem of functional analysis, Rec. Math. Moscou, n. Ser., Volume 4 (1938), pp. 471-497 | Zbl

[Ver81] Veretennikov, Alexander J. On strong solutions and explicit formulas for solutions of stochastic integral equations, Math. USSR, Sb., Volume 39 (1981) no. 3, pp. 387-403 | DOI | Zbl

[Xia97] Xiao, Yimin Hölder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields, Probab. Theory Relat. Fields, Volume 109 (1997) no. 1, pp. 129-157 | DOI | Zbl

[Zvo75] Zvonkin, Alexander K. A transformation of the phase space of a diffusion process that removes the drift, Math. USSR, Sb., Volume 22 (1974) (1975) no. 1, pp. 129-149 | Zbl